Problem 23 - Single Variable Calculus

Question

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\frac{2 n}{\sqrt{n+1}}\)

Step-by-Step Solution

Answer

The sequence \(a_n = \frac{2n}{\sqrt{n+1}}\) diverges as the limit as \(n\) approaches infinity is infinite: \[\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{2n}{\sqrt{n+1}} = \infty.\]

Step 1: Identify the given sequence

Consider the sequence \(a_n = \frac{2n}{\sqrt{n+1}}\) for \(n\in\mathbb{N}\).

Step 2: Recall the definition of convergence

A sequence \(\left\\{a_{n}\right\\}\) converges to a limit \(L\) if, as \(n\) approaches infinity, the terms of the sequence get arbitrarily close to \(L\). In other words, for every \(\varepsilon > 0\), there exists an integer \(N\) such that \(|a_n - L| < \varepsilon\) for all \(n > N\). If no such limit exists, the sequence diverges.

Step 3: Look for the limit of the sequence

To find the limit of the sequence as \(n\) approaches infinity, we use the following limit formula: \[\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{2n}{\sqrt{n+1}}.\]

Step 4: Divide the numerator and the denominator by the highest power of n in the denominator

In this case, the highest power of \(n\) in the denominator is \(n^{1/2}\), so we divide both the numerator and the denominator by \(n^{1/2}\): \[\lim_{n \to \infty} \frac{2n}{\sqrt{n+1}} = \lim_{n \to \infty} \frac{2n/n^{1/2}}{\sqrt{n+1}/n^{1/2}}.\] Simplify the expression: \[\lim_{n \to \infty} \frac{2\sqrt{n}}{\sqrt{\frac{n+1}{n}}}.\]

Step 5: Evaluate the limit

Now, we evaluate the limit as \(n\) goes to infinity: \[\lim_{n \to \infty} \frac{2\sqrt{n}}{\sqrt{\frac{n+1}{n}}} = \frac{2\sqrt{\infty}}{\sqrt{\frac{\infty + 1}{\infty}}}.\] We observe that \(\frac{\infty+1}{\infty}\) approaches 1, because the contribution of the constant term 1 becomes negligible compared to the term with \(n\) as \(n\) grows large. Thus, we have: \[\lim_{n \to \infty} \frac{2\sqrt{n}}{\sqrt{\frac{n+1}{n}}} = \frac{2\sqrt{\infty}}{\sqrt{1}}.\]

Step 6: Determine whether the sequence converges or diverges

Since \(\sqrt{\infty}\) is infinite, the limit of the sequence is also infinite: \[\lim_{n \to \infty} \frac{2\sqrt{n}}{\sqrt{\frac{n+1}{n}}} = \infty.\] Since the limit is infinite, the sequence \(\left\\{a_{n}\right\\}\) does not converge to a finite value; therefore, the sequence \(\left\\{a_{n}\right\\}\) diverges.

Key Concepts

Limit Evaluation

Evaluating the limit of a sequence is a fundamental aspect in understanding its behavior as it progresses towards infinity. To evaluate the limit of the given sequence \(a_n = \frac{2n}{\sqrt{n+1}}\), we follow specific mathematical procedures. These typically involve simplifying the fractional expression by dividing both the numerator and the denominator by the highest power of \(n\) found in the denominator.
In this example, we have \(n^{1/2}\) as the highest power, which simplifies the expression by separating \(n\) to isolate the leading terms in infinity. The simplification helps clarify how each component behaves as \(n\to\infty\), ultimately showing the limit as a representation of the sequence's long-term behavior.
It is crucial to approach these steps methodically, paying attention to how constants and contributing terms affect the final limit value.

Convergent and Divergent Sequences

Understanding whether a sequence is convergent or divergent is pivotal in mathematical analysis. Convergent sequences have a property where their terms approach a specific finite limit as \(n\) becomes infinitely large. Meanwhile, divergent sequences do not settle to a single finite limit— instead, they may diverge to infinity or fail to approach any particular value.
In the context of our example sequence \(a_n = \frac{2n}{\sqrt{n+1}}\), by assessing the limit and finding that it results in infinity, we determine that the sequence is divergent. Every time the magnitude of the sequence persistently increases without bound, it signifies divergence.
Recognizing these behaviors helps you predict and understand the trends within sequences and their potential impact on broader mathematical problems. This analysis feeds into broader questions, such as whether certain series based on such sequences are convergent or divergent.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers where each number corresponds to a specific position or index within that list. Sequences can be finite with a set number of terms, or infinite, potentially continuing indefinitely.
These sequences are defined by specific rules that generate successive terms, often based on a formula like \(a_n = \frac{2n}{\sqrt{n+1}}\). Studying these rules reveals not just the sequence's immediate behavior but its trends as \(n\) grows larger.
Working with sequences involves recognizing patterns, understanding growth rates, and evaluating limits, especially in infinite sequences. Mastery of these components applies to various fields from pure math to applied sciences. Such issues encompass estimating rates of change, determining bounds of functions, or assessing trends across numerous applications. Mathematical sequences present foundational concepts that anchor deeper exploration into series and broader calculus topics.